Annuity and Compound Interest Calculations

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Questions and Answers

What is the future value of an annuity due of $200 paid at the beginning of each week for 2 years in an account that pays 2.5% compounded weekly?

  • $43,177.12 (correct)
  • $42,463.66
  • $43,517.02
  • $42,798.73

Find the future value of an ordinary annuity where $600 is deposited at the end of each month for 5 years at 9% interest compounded monthly.

  • $43,517.02 (correct)
  • $42,798.73
  • $43,177.12
  • $42,463.66

What is the present value of an annuity of $7,500 paid at the end of each half-year for 10 years in an account bearing 9% compounded annually?

  • $68,531.08 (correct)
  • $69,152.21
  • $68,802.60
  • $69,599.30

If Sandra contributes GH630 at the end of each month to her retirement account that pays 8% compounded semi-annually, how much will she have when she retires 20 years from the start of contributions?

<p>GH194,126.33 (C)</p> Signup and view all the answers

Rosemary wants to buy a chalet for $300,000. She can save $10,000 a month in an account paying 15% interest compounded monthly. How long will it take her to save enough?

<p>3.2 years (C)</p> Signup and view all the answers

Jenny needs $20,000. She is saving $875 at the end of each month in an account bearing 15% interest compounded annually. When will the $20,000 be available?

<p>2.1 years (B)</p> Signup and view all the answers

Find the future fund for Kelly, who is saving $350 at the end of each month for the next 3 years, if her savings account bears 7% interest compounded quarterly.

<p>$13,438.71 (A)</p> Signup and view all the answers

What is the formula used to calculate the current value of a deferred annuity?

<p>𝐶𝑉𝑑𝑒𝑓 = 𝐴[1 − (1 + 𝑟)−(𝑛−𝑑) ]/𝑟 ( 1 + 𝑟)−𝑑 (A)</p> Signup and view all the answers

In the example provided, what is the value of 'A' when calculating the current value of an annuity due?

<p>800 (D)</p> Signup and view all the answers

What is the formula used to calculate the annual scholarship amount in the perpetuity example?

<p>𝐴 = 𝑟. 𝐶𝑉∞ (A)</p> Signup and view all the answers

What is the interest rate per period for an annuity that is compounded semiannually and pays a 8% annual interest?

<p>0.04 (A)</p> Signup and view all the answers

What is the formula used to calculate the present value of a perpetuity, which is the original family gift to the church in the 'Try' section?

<p>𝐶𝑉∞ = A/r (B)</p> Signup and view all the answers

In the example of setting up a scholarship fund, what does the value of $750,000 represent?

<p>Present value of the perpetuity (A)</p> Signup and view all the answers

What is the present value of an annuity due, given that the annuity payment is $900 each week for 1.5 years at 8% interest compounded weekly? Note that the interest rate given is an annual rate.

<p>8,108.51 (A)</p> Signup and view all the answers

In the 'Try' section, what is the present value of the perpetuity to generate a $2,650 monthly payment from the invested fund?

<p>$397,500 (C)</p> Signup and view all the answers

What is the formula used to calculate the current value of an annuity due?

<p>$CV_d = A \cdot a_{n,r} (1 + r)$ (C)</p> Signup and view all the answers

In the context of an annuity due, what does the term 'deferment period' refer to?

<p>The time period before the first payment is made (D)</p> Signup and view all the answers

If Robert and his wife deposit $6,600 annually for 5 years at a 7% interest rate, which of these calculates how much they will have at the end of the term?

<p>$FV_d = 6600 \cdot S_{5,0.07}$ (A)</p> Signup and view all the answers

What is the main characteristic of a deferred annuity?

<p>Payments start at a later date after the initial deposit period (B)</p> Signup and view all the answers

For an individual who deposits $550 monthly for 5 years at an interest rate of 7% compounded monthly, which approach gives the correct future value?

<p>$FV_d = 550 \cdot S_{60,0.07/12} (1 + 0.07/12)$ (D)</p> Signup and view all the answers

What is the impact of the compounding frequency on the current value of an annuity due?

<p>It affects the interest accrued on each payment (A)</p> Signup and view all the answers

If the interest rate is compounded semiannually, which of the following accurately describes how to adjust the calculations?

<p>The interest rate is halved and the number of periods is doubled (B)</p> Signup and view all the answers

In the context of calculating the future value of a deferred annuity, what does 'n' represent?

<p>The number of periods before the first payment (C)</p> Signup and view all the answers

Flashcards

Annuity Due

An annuity where payments occur at the beginning of each payment period. Examples include insurance premiums and property rentals.

Future Value of Annuity Due

The future value of an annuity due is calculated by multiplying the future value of an ordinary annuity by (1+r), where 'r' is the interest rate per period.

Present Value of Annuity Due

The present value of an annuity due is calculated by multiplying the present value of an ordinary annuity by (1+r), where 'r' is the interest rate per period.

Deferred Annuity

A type of annuity where payments are delayed. The payment period starts at a later point in time. For example, an annuity that starts in 5 years.

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Perpetuity

An annuity that continues forever. This is also known as a perpetual annuity.

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Present Value of a Perpetuity

The present value of a perpetuity is calculated by dividing the periodic payment by the interest rate. The calculation assumes that the interest rate remains constant.

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Annuity

A series of equal payments made over a defined period of time. Examples include monthly mortgage payments or regular contributions to a retirement account.

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Ordinary Annuity

An annuity where payments occur at the end of each payment period. Examples include monthly loan payments or regular savings deposits.

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Current Value of Annuity Due

The current value (present value) of an annuity due is the sum of the present values of all future payments, discounted back to the present time.

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Table Method for Annuities

The table method uses pre-calculated factors from financial tables to find the future and current values of annuities. These factors simplify the calculations.

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Future Value of Deferred Annuity

The future value of a deferred annuity is the total amount accumulated at the end of the annuity period, considering the deferment period. You essentially calculate the future value at the end of the deferment period and then use it to calculate the future value of the annuity.

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Current Value of Deferred Annuity

The current value of a deferred annuity is the present value of the future payments, discounted back to the present time, taking into account the deferment period.

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Deferment Period

The deferment period is the time between when the annuity is set up and when the first payment is made.

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Present Value of a Deferred Annuity

The present value of a deferred annuity is calculated by multiplying the present value of an ordinary annuity by (1 + r)^-d, where r is the interest rate per period and d is the deferment period. This formula helps to determine the current value of an annuity that starts in the future.

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Future Value of an Annuity Due

The future value of an annuity due is calculated by multiplying the future value of an ordinary annuity by (1 + r), where r is the interest rate per period. This formula assumes that payments are made at the beginning of each period, rather than at the end.

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What is the purpose of the formula 𝑎𝑛 ¬𝑟 (𝟏 + 𝒓)−𝒅?

This formula is used to calculate ________________. The 'a' represents the present value of an ordinary annuity, 'r' represents the interest rate per period, and 'd' represents the deferment period.

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What is the formula 𝐴 = 𝑟.𝐶𝑉∞ used for?

The formula 𝐴 = 𝑟.𝐶𝑉∞ is used to calculate the _______________. 'A' is the constant payment amount, 'r' is the interest rate per period, and 'CV∞' is the present value of the perpetuity. It represents the initial amount needed to sustain the perpetual payments.

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Study Notes

Actuarial Science Week 3 - Recap of Annuities

  • Annuities: Series of fixed payments made at regular intervals.
  • Types of Annuities: Ordinary annuity, Annuity due, Deferred annuity, Perpetuity.
  • Ordinary Annuity: Payments occur at the end of each period (e.g., month, quarter, year).
  • Annuity Due: Payments occur at the beginning of each period.
  • Deferred Annuity: Payoff starts at a future date. The first payment is made after a defined period (deferment period).
  • Perpetuity: An annuity that continues forever, with infinite payment periods.

Actuarial Science Week 3 - Calculations & Examples

  • Future Value (FV): The total value at a future date.
  • Future Value of an Ordinary Annuity: Formula (FV = A * [(1+r)^n - 1] / r); where A is periodic payment; r is periodic interest rate; n is number of periods
  • Future Value of an Annuity Due: Formula (FV_a = A * [((1+r)^n - 1) / r] * (1 + r) )
  • Current Value (CV): The present value of the annuity.
  • Current Value of an Ordinary Annuity: Formula (CV = A * [1 - (1 + r)^-n] / r); where A is periodic payment; r is periodic interest rate; n is number of periods.
  • Current Value of an Annuity Due: Formula (CV_a = A * [(1-(1+r)^-n)/r] * (1+r))
  • Table Method: Using present value tables or future value tables for calculations, avoiding direct formula application.
  • Compounding: Calculating interest on principal and accumulated interest.
  • Examples: Various scenarios illustrate annuity concept applications, including financial planning and investment.
  • Investment Problems: Determine future values for monthly or quarterly savings and the duration to reach a desired amount.

Actuarial Science Week 3 - Specialized Annuities

  • Deferred Annuity Future Value: Formula (FVdef = A * S_n¬r); where A is periodic payment; r is periodic interest rate; n is number of payment periods post-deferral (including deferral period). This is derived from future value of a regular annuity.
  • Deferred Annuity Current Value: Formula (CVdef = A * a_n¬r(1+r)^-d); where d is deferral period; A is periodic payment; r is periodic interest rate; n is number of periods, including the deferral period. Derived from the calculation for current value of ordinary annuity.
  • Perpetuity Formula: A = r * CV∞, where A is the annual payment, r is the interest rate, CV∞ is the present value of the perpetuity.

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